Subgroups — C₆₇×#61717 #30740968

Order 17152.

2014 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3354	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3353	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3332	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3330	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3229	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3226	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3194	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3101	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3100	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3092	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2975	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₆₇	—	2	1	no	—
15	#559×C₆₇	—	2	2	no	—
16	#556×C₆₇	C₂²	1	2	yes	complemented
17	#555×C₆₇	C₂²	1	2	yes	complemented
18	#553×C₆₇	—	2	2	no	—
19	#550×C₆₇	C₂²	1	2	yes	complemented
20	#547×C₆₇	—	2	2	no	—
21	#547×C₆₇	—	2	2	no	—
22	#547×C₆₇	C₂²	1	4	yes	—
23	#547×C₆₇	C₂²	1	4	yes	—
24	#547×C₆₇	C₂²	1	2	yes	complemented
25	#538×C₆₇	—	2	2	no	—
26	#535×C₆₇	—	2	1	no	—
27	#533×C₆₇	C₂²	1	1	yes	characteristic
28	#532×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	#525×C₆₇	C₂²	1	4	yes	—
31	C₂×#123×C₆₇	—	2	2	no	—
32	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
33	C₂×#121×C₆₇	C₂²	1	2	yes	—
34	#520×C₆₇	—	2	1	no	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	#500×C₆₇	C₂²	1	2	yes	complemented
39	#498×C₆₇	C₂²	1	2	yes	complemented
40	#487×C₆₇	C₂²	1	2	yes	complemented
41	#486×C₆₇	C₂²	1	2	yes	complemented
42	#484×C₆₇	C₂²	1	8	yes	complemented
43	#483×C₆₇	C₂²	1	2	yes	—
44	#483×C₆₇	C₂²	1	4	yes	—
45	#482×C₆₇	C₂²	1	8	yes	complemented
46	#481×C₆₇	C₂²	1	8	yes	complemented
47	#480×C₆₇	C₂²	1	2	yes	—
48	#480×C₆₇	C₂²	1	4	yes	—
49	#476×C₆₇	C₂²	1	2	yes	complemented
50	#475×C₆₇	C₂²	1	4	yes	complemented
51	#474×C₆₇	C₂²	1	4	yes	complemented
52	#474×C₆₇	C₂²	1	2	yes	complemented
53	#469×C₆₇	C₂²	1	8	yes	complemented
54	#468×C₆₇	C₂²	1	8	yes	complemented
55	#461×C₆₇	C₂²	1	2	yes	complemented
56	#459×C₆₇	C₂²	1	2	yes	complemented
57	#450×C₆₇	C₂²	1	2	yes	—
58	#449×C₆₇	C₂²	1	8	yes	complemented
59	#447×C₆₇	C₂²	1	2	yes	—
60	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
61	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
62	#436×C₆₇	C₂²	1	4	yes	complemented
63	#436×C₆₇	C₂²	1	4	yes	complemented
64	#431×C₆₇	C₂²	1	1	yes	characteristic
65	#431×C₆₇	C₂²	1	1	yes	characteristic
66	#428×C₆₇	C₂²	1	8	yes	complemented
67	#417×C₆₇	C₂²	1	8	yes	complemented
68	#407×C₆₇	C₂²	1	4	yes	—
69	C₂×#41×C₆₇	—	2	4	no	—
70	C₂×#41×C₆₇	—	2	4	no	—
71	C₂×#41×C₆₇	—	2	1	no	—
72	C₂×#41×C₆₇	—	4	8	no	—
73	C₂×#41×C₆₇	C₂³	1	4	yes	—
74	C₂²×Q₈×C₆₇	—	2	1	no	—
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	—	2	2	no	—
79	C₂³×C₄×C₆₇	—	2	1	no	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
82	C₂×D₈×C₆₇	C₂³	1	4	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
84	C₄⋊Q₈×C₆₇	—	2	1	no	—
85	C₄⋊Q₈×C₆₇	—	2	1	no	—
86	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
88	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
91	#127×C₆₇	—	2	2	no	—
92	#127×C₆₇	C₂³	1	1	yes	characteristic
93	#126×C₆₇	—	2	4	no	—
94	#125×C₆₇	—	2	1	no	—
95	#124×C₆₇	—	2	2	no	—
96	#124×C₆₇	—	2	4	no	—
97	#124×C₆₇	—	2	1	no	—
98	#123×C₆₇	—	2	4	no	—
99	#123×C₆₇	—	2	8	no	—
100	#123×C₆₇	—	2	8	no	—
101	#123×C₆₇	—	2	4	no	—
102	C₂²⋊Q₈×C₆₇	—	2	1	no	—
103	C₂²⋊Q₈×C₆₇	—	2	4	no	—
104	C₂²⋊Q₈×C₆₇	—	2	4	no	—
105	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
106	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
107	#121×C₆₇	—	2	2	no	—
108	#121×C₆₇	—	2	2	no	—
109	#121×C₆₇	—	2	4	no	—
110	#121×C₆₇	—	2	1	no	—
111	#121×C₆₇	—	2	4	no	—
112	#121×C₆₇	C₂³	1	8	yes	—
113	#121×C₆₇	C₂³	1	4	yes	—
114	#121×C₆₇	C₂³	1	2	yes	—
115	#120×C₆₇	—	2	4	no	—
116	C₄×Q₈×C₆₇	—	2	2	no	—
117	C₄×Q₈×C₆₇	—	2	1	no	—
118	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
119	C₄×D₄×C₆₇	—	2	4	no	—
120	C₄×D₄×C₆₇	—	2	4	no	—
121	C₄×D₄×C₆₇	—	2	2	no	—
122	C₄×D₄×C₆₇	—	2	1	no	—
123	C₄×D₄×C₆₇	—	2	1	no	—
124	C₄×D₄×C₆₇	—	2	1	no	—
125	C₄×D₄×C₆₇	D₄	1	2	yes	—
126	C₄×D₄×C₆₇	D₄	1	2	yes	—
127	C₄×D₄×C₆₇	C₂³	1	2	yes	—
128	C₄×D₄×C₆₇	C₂³	1	2	yes	—
129	C₄×D₄×C₆₇	C₂³	1	2	yes	—
130	C₄×D₄×C₆₇	C₂³	1	2	yes	—
131	C₄×D₄×C₆₇	C₂³	1	2	yes	—
132	#117×C₆₇	—	2	1	no	—
133	#117×C₆₇	—	2	2	no	—
134	#117×C₆₇	C₂³	1	4	yes	—
135	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
137	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
140	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
143	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
144	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
146	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
147	#107×C₆₇	C₂³	1	8	yes	—
148	#107×C₆₇	C₂³	1	4	yes	—
149	#106×C₆₇	C₂³	1	8	yes	—
150	#106×C₆₇	C₂³	1	4	yes	—
151	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
152	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
153	#103×C₆₇	C₂³	1	8	yes	—
154	#103×C₆₇	C₂³	1	4	yes	—
155	#102×C₆₇	C₂³	1	8	yes	—
156	#102×C₆₇	C₂³	1	4	yes	—
157	#102×C₆₇	C₂³	1	4	yes	—
158	#102×C₆₇	C₂³	1	4	yes	—
159	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
160	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
161	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
162	C₄×C₈×C₆₇	C₂³	1	2	yes	—
163	C₄×C₈×C₆₇	C₂³	1	2	yes	—
164	C₂⁴×C₆₇	—	2	1	no	—
165	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
166	#41×C₆₇	—	4	8	no	—
167	#41×C₆₇	—	4	16	no	—
168	#41×C₆₇	—	4	16	no	—
169	#41×C₆₇	—	4	4	no	—
170	C₂×Q₈×C₆₇	—	2	1	no	—
171	C₂×Q₈×C₆₇	—	2	2	no	—
172	C₂×Q₈×C₆₇	—	2	2	no	—
173	C₂×Q₈×C₆₇	—	2	1	no	—
174	C₂×Q₈×C₆₇	—	4	4	no	—
175	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
176	C₂×D₄×C₆₇	—	2	2	no	—
177	C₂×D₄×C₆₇	—	2	8	no	—
178	C₂×D₄×C₆₇	—	2	4	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	2	1	no	—
182	C₂×D₄×C₆₇	—	2	1	no	—
183	C₂×D₄×C₆₇	—	2	1	no	—
184	C₂×D₄×C₆₇	—	4	2	no	—
185	C₂×D₄×C₆₇	—	4	8	no	—
186	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
187	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
188	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
189	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₆₇	—	2	1	no	—
191	C₂²×C₄×C₆₇	—	2	2	no	—
192	C₂²×C₄×C₆₇	—	2	8	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	2	no	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	4	4	no	—
198	C₂²×C₄×C₆₇	—	4	8	no	—
199	C₂²×C₄×C₆₇	—	4	4	no	—
200	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
203	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
206	SD₁₆×C₆₇	—	4	4	no	—
207	D₈×C₆₇	—	4	4	no	—
208	M₁₆×C₆₇	—	4	8	no	—
209	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
210	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
211	C₄⋊C₄×C₆₇	—	2	1	no	—
212	C₄⋊C₄×C₆₇	—	2	2	no	—
213	C₄⋊C₄×C₆₇	—	2	8	no	—
214	C₄⋊C₄×C₆₇	—	2	1	no	—
215	C₄⋊C₄×C₆₇	—	2	4	no	—
216	C₄⋊C₄×C₆₇	—	2	2	no	—
217	C₄⋊C₄×C₆₇	—	2	2	no	—
218	C₄⋊C₄×C₆₇	—	2	1	no	—
219	C₄⋊C₄×C₆₇	—	2	2	no	—
220	C₄⋊C₄×C₆₇	—	2	1	no	—
221	C₄⋊C₄×C₆₇	—	2	1	no	—
222	C₄⋊C₄×C₆₇	—	2	2	no	—
223	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
224	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
226	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
227	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
228	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
229	C₂²⋊C₄×C₆₇	—	2	2	no	—
230	C₂²⋊C₄×C₆₇	—	2	4	no	—
231	C₂²⋊C₄×C₆₇	—	2	4	no	—
232	C₂²⋊C₄×C₆₇	—	2	2	no	—
233	C₂²⋊C₄×C₆₇	—	2	2	no	—
234	C₂²⋊C₄×C₆₇	—	2	8	no	—
235	C₂²⋊C₄×C₆₇	—	2	4	no	—
236	C₂²⋊C₄×C₆₇	—	2	2	no	—
237	C₂²⋊C₄×C₆₇	—	2	2	no	—
238	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
239	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
240	C₄²×C₆₇	—	2	4	no	—
241	C₄²×C₆₇	—	2	2	no	—
242	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
243	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
244	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
245	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
246	C₂³×C₆₇	—	2	4	no	—
247	C₂³×C₆₇	—	2	2	no	—
248	C₂³×C₆₇	—	2	2	no	—
249	C₂³×C₆₇	—	4	2	no	—
250	C₂³×C₆₇	—	4	2	no	—
251	C₂³×C₆₇	—	4	2	no	—
252	C₂³×C₆₇	—	4	2	no	—
253	C₂³×C₆₇	—	4	2	no	—
254	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
255	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	Q₈×C₆₇	—	2	4	no	—
257	Q₈×C₆₇	—	4	4	no	—
258	Q₈×C₆₇	—	4	4	no	—
259	Q₈×C₆₇	—	4	2	no	—
260	Q₈×C₆₇	—	8	1	no	—
261	D₄×C₆₇	—	2	4	no	—
262	D₄×C₆₇	—	2	4	no	—
263	D₄×C₆₇	—	2	2	no	—
264	D₄×C₆₇	—	2	4	no	—
265	D₄×C₆₇	—	2	2	no	—
266	D₄×C₆₇	—	4	2	no	—
267	D₄×C₆₇	—	4	16	no	—
268	D₄×C₆₇	—	4	4	no	—
269	D₄×C₆₇	—	4	2	no	—
270	D₄×C₆₇	—	8	1	no	—
271	D₄×C₆₇	—	8	4	no	—
272	D₄×C₆₇	—	8	4	no	—
273	C₂×C₄×C₆₇	—	2	8	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	4	no	—
280	C₂×C₄×C₆₇	—	4	4	no	—
281	C₂×C₄×C₆₇	—	4	2	no	—
282	C₂×C₄×C₆₇	—	4	1	no	—
283	C₂×C₄×C₆₇	—	4	16	no	—
284	C₂×C₄×C₆₇	—	4	2	no	—
285	C₂×C₄×C₆₇	—	4	4	no	—
286	C₂×C₄×C₆₇	—	4	4	no	—
287	C₂×C₄×C₆₇	—	4	4	no	—
288	C₂×C₄×C₆₇	—	4	4	no	—
289	C₂×C₄×C₆₇	—	8	4	no	—
290	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
291	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
292	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
298	C₈×C₆₇	—	2	4	no	—
299	C₈×C₆₇	—	2	4	no	—
300	C₂²×C₆₇	—	2	1	no	—
301	C₂²×C₆₇	—	2	1	no	—
302	C₂²×C₆₇	—	2	4	no	—
303	C₂²×C₆₇	—	2	4	no	—
304	C₂²×C₆₇	—	2	4	no	—
305	C₂²×C₆₇	—	4	8	no	—
306	C₂²×C₆₇	—	4	2	no	—
307	C₂²×C₆₇	—	4	2	no	—
308	C₂²×C₆₇	—	4	2	no	—
309	C₂²×C₆₇	—	4	2	no	—
310	C₂²×C₆₇	—	4	2	no	—
311	C₂²×C₆₇	—	4	2	no	—
312	C₂²×C₆₇	—	8	4	no	—
313	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
314	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
315	C₄×C₆₇	—	2	1	no	—
316	C₄×C₆₇	—	2	1	no	—
317	C₄×C₆₇	—	2	1	no	—
318	C₄×C₆₇	—	2	4	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	2	no	—
322	C₄×C₆₇	—	4	4	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
325	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
326	C₄×C₆₇	#585	1	2	yes	—
327	#61717	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
328	C₂×C₆₇	—	2	2	no	—
329	C₂×C₆₇	—	4	4	no	—
330	C₂×C₆₇	—	8	2	no	—
331	C₂×C₆₇	—	8	2	no	—
332	C₂×C₆₇	#3485	1	1	yes	characteristic
333	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂×C₆₇	#3586	1	1	yes	characteristic
335	#3506	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₂×#547	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#3354	C₂×C₆₇	1	4	yes	complemented
338	#3353	C₂×C₆₇	1	4	yes	complemented
339	#3332	C₂×C₆₇	1	4	yes	complemented
340	#3330	C₂×C₆₇	1	4	yes	complemented
341	#3229	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#3226	C₂×C₆₇	1	1	yes	characteristic, complemented
343	#3194	C₂×C₆₇	1	4	yes	complemented
344	#3101	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#3100	C₂×C₆₇	1	1	yes	characteristic, complemented
346	#3092	C₂×C₆₇	1	4	yes	complemented
347	#2975	C₂×C₆₇	1	1	yes	characteristic, complemented
348	C₆₇	#61717	1	1	yes	characteristic, direct product factor, Sylow
349	C₂²×#41	—	2	1	no	—
350	#559	—	2	2	no	—
351	#556	C₂²×C₆₇	1	2	yes	complemented
352	#555	C₂²×C₆₇	1	2	yes	complemented
353	#553	—	2	2	no	—
354	#550	C₂²×C₆₇	1	2	yes	complemented
355	#547	—	2	2	no	—
356	#547	—	2	2	no	—
357	#547	C₂²×C₆₇	1	4	yes	—
358	#547	C₂²×C₆₇	1	4	yes	—
359	#547	C₂²×C₆₇	1	2	yes	complemented
360	#538	—	2	2	no	—
361	#535	—	2	1	no	—
362	#533	C₂²×C₆₇	1	1	yes	characteristic
363	#532	C₂²×C₆₇	1	1	yes	characteristic
364	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
365	#525	C₂²×C₆₇	1	4	yes	—
366	C₂×#123	—	2	2	no	—
367	C₂×C₂²⋊Q₈	—	2	1	no	—
368	C₂×#121	C₂²×C₆₇	1	2	yes	—
369	#520	—	2	1	no	—
370	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
371	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
372	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
373	#500	C₂²×C₆₇	1	2	yes	complemented
374	#498	C₂²×C₆₇	1	2	yes	complemented
375	#487	C₂²×C₆₇	1	2	yes	complemented
376	#486	C₂²×C₆₇	1	2	yes	complemented
377	#484	C₂²×C₆₇	1	8	yes	complemented
378	#483	C₂²×C₆₇	1	2	yes	—
379	#483	C₂²×C₆₇	1	4	yes	—
380	#482	C₂²×C₆₇	1	8	yes	complemented
381	#481	C₂²×C₆₇	1	8	yes	complemented
382	#480	C₂²×C₆₇	1	2	yes	—
383	#480	C₂²×C₆₇	1	4	yes	—
384	#476	C₂²×C₆₇	1	2	yes	complemented
385	#475	C₂²×C₆₇	1	4	yes	complemented
386	#474	C₂²×C₆₇	1	4	yes	complemented
387	#474	C₂²×C₆₇	1	2	yes	complemented
388	#469	C₂²×C₆₇	1	8	yes	complemented
389	#468	C₂²×C₆₇	1	8	yes	complemented
390	#461	C₂²×C₆₇	1	2	yes	complemented
391	#459	C₂²×C₆₇	1	2	yes	complemented
392	#450	C₂²×C₆₇	1	2	yes	—
393	#449	C₂²×C₆₇	1	8	yes	complemented
394	#447	C₂²×C₆₇	1	2	yes	—
395	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
396	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
397	#436	C₂²×C₆₇	1	4	yes	complemented
398	#436	C₂²×C₆₇	1	4	yes	complemented
399	#431	C₂²×C₆₇	1	1	yes	characteristic
400	#431	C₂²×C₆₇	1	1	yes	characteristic
401	#428	C₂²×C₆₇	1	8	yes	complemented
402	#417	C₂²×C₆₇	1	8	yes	complemented
403	#407	C₂²×C₆₇	1	4	yes	—
404	C₂×#41	—	2	4	no	—
405	C₂×#41	—	2	4	no	—
406	C₂×#41	—	2	1	no	—
407	C₂×#41	—	4	8	no	—
408	C₂×#41	C₂³×C₆₇	1	4	yes	—
409	C₂²×Q₈	—	2	1	no	—
410	C₂²×D₄	—	2	2	no	—
411	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
412	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂³×C₄	—	2	2	no	—
414	C₂³×C₄	—	2	1	no	—
415	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
417	C₂×D₈	C₂³×C₆₇	1	4	yes	—
418	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
419	C₄⋊Q₈	—	2	1	no	—
420	C₄⋊Q₈	—	2	1	no	—
421	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
422	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
423	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
424	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
425	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
426	#127	—	2	2	no	—
427	#127	C₂³×C₆₇	1	1	yes	characteristic
428	#126	—	2	4	no	—
429	#125	—	2	1	no	—
430	#124	—	2	2	no	—
431	#124	—	2	4	no	—
432	#124	—	2	1	no	—
433	#123	—	2	4	no	—
434	#123	—	2	8	no	—
435	#123	—	2	8	no	—
436	#123	—	2	4	no	—
437	C₂²⋊Q₈	—	2	1	no	—
438	C₂²⋊Q₈	—	2	4	no	—
439	C₂²⋊Q₈	—	2	4	no	—
440	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
441	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
442	#121	—	2	2	no	—
443	#121	—	2	2	no	—
444	#121	—	2	4	no	—
445	#121	—	2	1	no	—
446	#121	—	2	4	no	—
447	#121	C₂³×C₆₇	1	8	yes	—
448	#121	C₂³×C₆₇	1	2	yes	—
449	#121	C₂³×C₆₇	1	4	yes	—
450	#120	—	2	4	no	—
451	C₄×Q₈	—	2	2	no	—
452	C₄×Q₈	—	2	1	no	—
453	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
454	C₄×D₄	—	2	4	no	—
455	C₄×D₄	—	2	4	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	—	2	1	no	—
459	C₄×D₄	—	2	1	no	—
460	C₄×D₄	D₄×C₆₇	1	2	yes	—
461	C₄×D₄	D₄×C₆₇	1	2	yes	—
462	C₄×D₄	C₂³×C₆₇	1	2	yes	—
463	C₄×D₄	C₂³×C₆₇	1	2	yes	—
464	C₄×D₄	C₂³×C₆₇	1	2	yes	—
465	C₄×D₄	C₂³×C₆₇	1	2	yes	—
466	C₄×D₄	C₂³×C₆₇	1	2	yes	—
467	#117	—	2	1	no	—
468	#117	—	2	2	no	—
469	#117	C₂³×C₆₇	1	4	yes	—
470	C₂×C₄⋊C₄	—	2	2	no	—
471	C₂×C₄⋊C₄	—	2	4	no	—
472	C₂×C₄⋊C₄	—	2	1	no	—
473	C₂×C₄⋊C₄	—	2	2	no	—
474	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
475	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
476	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
477	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₂²⋊C₄	—	2	2	no	—
480	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
481	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
482	#107	C₂³×C₆₇	1	4	yes	—
483	#107	C₂³×C₆₇	1	8	yes	—
484	#106	C₂³×C₆₇	1	4	yes	—
485	#106	C₂³×C₆₇	1	8	yes	—
486	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
487	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
488	#103	C₂³×C₆₇	1	4	yes	—
489	#103	C₂³×C₆₇	1	8	yes	—
490	#102	C₂³×C₆₇	1	4	yes	—
491	#102	C₂³×C₆₇	1	8	yes	—
492	#102	C₂³×C₆₇	1	4	yes	—
493	#102	C₂³×C₆₇	1	4	yes	—
494	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
495	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
496	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
497	C₄×C₈	C₂³×C₆₇	1	2	yes	—
498	C₄×C₈	C₂³×C₆₇	1	2	yes	—
499	C₂⁴	—	2	1	no	—
500	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
501	#41	—	4	8	no	—
502	#41	—	4	16	no	—
503	#41	—	4	16	no	—
504	#41	—	4	4	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	2	2	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	4	4	no	—
510	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	8	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	—	2	4	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	1	no	—
518	C₂×D₄	—	2	1	no	—
519	C₂×D₄	—	4	2	no	—
520	C₂×D₄	—	4	8	no	—
521	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
522	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
523	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
524	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	8	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	4	no	—
532	C₂²×C₄	—	4	4	no	—
533	C₂²×C₄	—	4	8	no	—
534	C₂²×C₄	—	4	4	no	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
536	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
537	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
538	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
539	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
540	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
541	SD₁₆	—	4	4	no	—
542	D₈	—	4	4	no	—
543	M₁₆	—	4	8	no	—
544	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
545	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	8	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
559	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
560	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
561	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
562	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
563	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
564	C₂²⋊C₄	—	2	2	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₂²⋊C₄	—	2	8	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
574	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
575	C₄²	—	2	4	no	—
576	C₄²	—	2	2	no	—
577	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
578	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
579	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
580	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
581	C₂³	—	2	4	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	4	2	no	—
585	C₂³	—	4	2	no	—
586	C₂³	—	4	2	no	—
587	C₂³	—	4	2	no	—
588	C₂³	—	4	2	no	—
589	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
590	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
591	Q₈	—	2	4	no	—
592	Q₈	—	4	4	no	—
593	Q₈	—	4	4	no	—
594	Q₈	—	4	2	no	—
595	Q₈	—	8	1	no	—
596	D₄	—	2	4	no	—
597	D₄	—	2	4	no	—
598	D₄	—	2	2	no	—
599	D₄	—	2	4	no	—
600	D₄	—	2	2	no	—
601	D₄	—	4	2	no	—
602	D₄	—	4	16	no	—
603	D₄	—	4	4	no	—
604	D₄	—	4	2	no	—
605	D₄	—	8	1	no	—
606	D₄	—	8	4	no	—
607	D₄	—	8	4	no	—
608	C₂×C₄	—	2	8	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	16	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	4	no	—
621	C₂×C₄	—	4	4	no	—
622	C₂×C₄	—	4	4	no	—
623	C₂×C₄	—	4	4	no	—
624	C₂×C₄	—	8	4	no	—
625	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
626	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
627	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
629	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
630	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
631	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
632	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
633	C₈	—	2	4	no	—
634	C₈	—	2	4	no	—
635	C₂²	—	2	1	no	—
636	C₂²	—	2	1	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	4	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	4	8	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	8	4	no	—
648	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
649	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
650	C₄	—	2	1	no	—
651	C₄	—	2	1	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	4	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	4	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
660	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
661	C₄	#585×C₆₇	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	4	no	—
664	C₂	—	8	2	no	—
665	C₂	—	8	2	no	—
666	C₂	C₆₇×#3485	1	1	yes	characteristic
667	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
668	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3506	C₂	semidirect
2. C₂×#547×C₆₇	C₂	semidirect
3. C₆₇×#3354	C₂	semidirect
4. C₆₇×#3353	C₂	semidirect
5. C₆₇×#3332	C₂	semidirect
6. C₆₇×#3330	C₂	semidirect
7. C₆₇×#3229	C₂	semidirect
8. C₆₇×#3226	C₂	semidirect
9. C₆₇×#3194	C₂	semidirect
10. C₆₇×#3101	C₂	semidirect
11. C₆₇×#3100	C₂	semidirect
12. C₆₇×#3092	C₂	semidirect
13. C₆₇×#2975	C₂	semidirect
16. #556×C₆₇	C₂²	semidirect
17. #555×C₆₇	C₂²	semidirect
19. #550×C₆₇	C₂²	semidirect
24. #547×C₆₇	C₂²	semidirect
38. #500×C₆₇	C₂²	semidirect
39. #498×C₆₇	C₂²	semidirect
40. #487×C₆₇	C₂²	semidirect
41. #486×C₆₇	C₂²	semidirect
42. #484×C₆₇	C₂²	semidirect
45. #482×C₆₇	C₂²	semidirect
46. #481×C₆₇	C₂²	semidirect
49. #476×C₆₇	C₂²	semidirect
50. #475×C₆₇	C₂²	semidirect
51. #474×C₆₇	C₂²	semidirect
52. #474×C₆₇	C₂²	semidirect
53. #469×C₆₇	C₂²	semidirect
54. #468×C₆₇	C₂²	semidirect
55. #461×C₆₇	C₂²	semidirect
56. #459×C₆₇	C₂²	semidirect
58. #449×C₆₇	C₂²	semidirect
60. C₄×SD₁₆×C₆₇	C₂²	semidirect
61. C₄×D₈×C₆₇	C₂²	semidirect
62. #436×C₆₇	C₂²	semidirect
63. #436×C₆₇	C₂²	semidirect
66. #428×C₆₇	C₂²	semidirect
67. #417×C₆₇	C₂²	semidirect
327. #61717	C₆₇	direct (N × H)
335. #3506	C₂×C₆₇	semidirect
336. C₂×#547	C₂×C₆₇	semidirect
337. #3354	C₂×C₆₇	semidirect
338. #3353	C₂×C₆₇	semidirect
339. #3332	C₂×C₆₇	semidirect
340. #3330	C₂×C₆₇	semidirect
341. #3229	C₂×C₆₇	semidirect
342. #3226	C₂×C₆₇	semidirect
343. #3194	C₂×C₆₇	semidirect
344. #3101	C₂×C₆₇	semidirect
345. #3100	C₂×C₆₇	semidirect
346. #3092	C₂×C₆₇	semidirect
347. #2975	C₂×C₆₇	semidirect
348. C₆₇	#61717	direct (N × H)
351. #556	C₂²×C₆₇	semidirect
352. #555	C₂²×C₆₇	semidirect
354. #550	C₂²×C₆₇	semidirect
359. #547	C₂²×C₆₇	semidirect
373. #500	C₂²×C₆₇	semidirect
374. #498	C₂²×C₆₇	semidirect
375. #487	C₂²×C₆₇	semidirect
376. #486	C₂²×C₆₇	semidirect
377. #484	C₂²×C₆₇	semidirect
380. #482	C₂²×C₆₇	semidirect
381. #481	C₂²×C₆₇	semidirect
384. #476	C₂²×C₆₇	semidirect
385. #475	C₂²×C₆₇	semidirect
386. #474	C₂²×C₆₇	semidirect
387. #474	C₂²×C₆₇	semidirect
388. #469	C₂²×C₆₇	semidirect
389. #468	C₂²×C₆₇	semidirect
390. #461	C₂²×C₆₇	semidirect
391. #459	C₂²×C₆₇	semidirect
393. #449	C₂²×C₆₇	semidirect
395. C₄×SD₁₆	C₂²×C₆₇	semidirect
396. C₄×D₈	C₂²×C₆₇	semidirect
397. #436	C₂²×C₆₇	semidirect
398. #436	C₂²×C₆₇	semidirect
401. #428	C₂²×C₆₇	semidirect
402. #417	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3506	2
C₂×#547×C₆₇	2
C₆₇×#3354	2
C₆₇×#3353	2
C₆₇×#3332	2
C₆₇×#3330	2
C₆₇×#3229	2
C₆₇×#3226	2
C₆₇×#3194	2
C₆₇×#3101	2
C₆₇×#3100	2
C₆₇×#3092	2
C₆₇×#2975	2
#61717	67

Subgroups — C67×#61717 #30740968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61717 #30740968